[Notes]

During my PhD study, I've been questioning about, what is re-search.

It seems to me that research is about

• looking at something,
• over and over,
• in all possible angels.

During these iterations, we may

• narrow assumptions,
• apply different methods,
• collect more data,
• and learn new knowledge.

• Reproducibility => why I present in this transparent way
• so it is informative about what I conclude and how I deliver

---

Over the past years, I studied about

• Globular Clusters, these are
  • stellar systems that consist of millions of stars, they are also
  • ancient building blocks of galaxies themselfs.

They are extremely interesting environments, in a sense that,

• the timescale of stellar evolution, becomes comparable to
• the timescale to have stellar encounters.

That is, the stars in these dense star clusters don't die alone,

• they will influence the nearby-stars over dynamical interactions,
• their own stellar evolutions are also altered during such processes.

In a nutshell,

• smaller systems like open clusters cannot stay gravitationally-binded
  • stars are easily scattered away and they will evaporate before long
  • there are also enough materials to power multiple generation of star formationss
• larger systems like galaxies instead, have strong gravitational potentials
  • most of the stars are trapped and move under a quasi-static field
  • the system could be described as statistical equilibrium,

Starting from that, I learned about

• the larger structure in the local universe and
• built up the concept of the cosmic web

I will expand on that by populating globular clusters in the local universe, which is the second chapter in my dissertation.

---

Then I had this chance to study about numerical simulations.

• I spent the 2013 summer in National Astronomical Observatories, Chinese Academy of Science to study about N-body models [with Rainer Spurzem], using Newtonian integrator to solve dynamical interactions.
• I also [met with Jonathon Downing and] picked up the Monte Carlo code [from Mirek Giersz] to evaluate the phase-space evolution in a simi-equilibrium system.

Both approaches provide great details about stellar dynamics based on theoretical assumptions.

• From the formation mechanisim of compact binary systems,
• to the observable global effects of the star clusters, different physics and phenomenons are implemented in various scale.

The adaptions of

• these numerical methods,
• in together with high performance computing techniques, push forward the research greatly so that theoretical prospects become
• essential to new proposed instrumental projects,
• instructional to reveal discoveries buried in observed data.

I reviewed such aspects in the history of GC studies in chapter 3 and introduced my own simulations to represent the natural assembling of GCs in chapter 4.

• The major effects considered in GCs and
• the detailed modeling are discussed in my dissertation in corresponding sections. I will only briefly demonstrate the key components later.

---

From those simulations, I could extract any interesting stellar object with its full dynamical history, espectially the compact binary systems that could be potentially the gravitational wave sources in different frequency domain.

• I spent 2015 summer at Kavli Institute of Astrophysics and Astronomy in Peking University doing data analysis by querying observational databases for variable sources,
• I visited National Astronomical Observatories, Chinese Academy of Science again to prototype a transient object detection pipeline for 50cm Binocular Network.

The mutual benefits supplimented my experience in optical data processing that promoted my understanding on time series data analysis.

I sketched the work flow to estimate the local abundance of GCs in chapter 2. At the same time, I started to focuz on the binary black holes for the resulting GW signals. Matt's code based on Peter's work about general relativity was parallelized to carry out the relativistic evolution for the compact binary systems extracted from GCs. A brief introduction about BBH formation mechanisms and GW astronomy could be found in chapter 5. At the time of the first GW detection, my preliminary result presented on MODEST meeting ruled out GC as the origin of the IMBH merger event.

---

In 2016 summer, I started training in the LSST data science program and

• the systematic coverage of statistical background
• and conventional data analysis tools enhanced my modeling of the GC populations.

With the supercluster in Texas Advanced Computing Center, my collection of GC simulations was also enlarged greatly.

• More realizations of the same setup were added to make sure the convergence.
• Finer-grained parameter space was explored to avoid extremity.
• Selection effect and sampling bias had been made aware as well.

---

From all the above, a clear image appeared

• to derive the detection prospects of BBHs,
• which are dynamically formed in GCs,
• within 30 Mpc volumetric space
• for space-borne GW detectors

I followed

• the Fixer Informative Matrix analysis from Cutler and
• configured the BBH database to work with Matt's code

This part of work is included in chapter 6.

---

Without further ado, I will walk through the work manuscripted in my dissertation in brief and let you question about any part that is particular interesting to your extent.

Prospects for observing dynamically formed Binary Black Holes in the Local Universe with Gravitational Waves

Dongming Jin

• Research Supervisor
  • Dr. Matthew Benacquista
• Committee Members
  • Dr. Zdzislaw Musielak (co-chair)
  • Dr. Manfred Cuntz
  • Dr. Joseph Romano
  • Dr. Sangwook Park
  • Dr. Teviet Creighton

• Graduate Advisor

  • Dr. Qiming Zhang

• April 27th, 2018

Outline

Introduction (chap 1)

Populating GCs in the local universe (chap 2)

Simulating GCs (chap 3 + chap 4)

Relativistic BBHs (chap 5)

Prospects for Detection (chap 6)

Introduction (chap 1)

Motivation

• Detection -> GWs from BBHs, overview about GW, BBH
• BBH formation -> GCs
• Research interns
  • N-body, MOCCA: Newtonian, statistical
  • 50BiN: optical, time series
  • DSFP: statistics, data analysis

Objective

Background

Introduction (chap 1)

Motivation

Objective

• GW from BBH formed in GCs -> Astrometry
  • High performance computing: numerical simulations on superclusters
  • Long baseline/timeline observation: time series analysis + multi-messanger
  • Petascale data: systematic surveys, global multi-discipline collaborations

Background

Introduction (chap 1)

Motivation

Objective

Background

• Astrometry (sec 2.1.2)
  • Cosmology (sec 1.2.1)
  • Dark matter content (sec 6.2.2)
• GCs and BBHs
  • GCs (sec 3.3)
  • BBHs (sec 5.2.2)
• GR and GWs from BBHs
  • GR -> GW (sec 5.1.2)
  • GW from BBH (sec 5.2.3)

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1)

Harris catalog for GCpG model (sec 2.2)

GCpG models (sec 2.2)

Modeling GC populations (sec 2.3)

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1)

• White et al. (2011)
• data source

Data prepare

1. [=] load GWGC data: wrong data type object due to ~ symbol => impute ~ as NaN and parse data type to float

2. [=] check duplicates: explain pandas table, duplicated galaxies with Dist > 30 Mpc => drop

3. extract local galaxies within 30 Mpc and 3D view

4. spatial distribution and edge galaxies

5. check interesting features

Description

   1-  7  I7    ---      PGC      [2,4715229]? Identifier from HYPERLEDA
                                  (empty for globular clusters)
   9- 36  A28   ---      Name     Common name of galaxy or globular
  38- 46  F9.5  h        RAhour   Right ascension (J2000, decimal hours)
  48- 55  F8.4  deg      DEdeg    Declination (J2000)
  57- 60  F4.1  ---      TT       [-9,10]? Morphological type code (1)
  62- 66  F5.2  mag      Bmag     ? Apparent blue magnitude
  68- 74  F7.3  arcmin   a        ? Major diameter (arcmins)
  76- 82  F7.3  arcmin   e_a      ? Error in major diameter (arcmins)
  84- 90  F7.3  arcmin   b        ? Minor diameter (arcmins)
  92- 98  F7.3  arcmin   e_b      ? Error in minor diameter (arcmins)
 100-104  F5.3  ---      b/a      [0,1]? Ratio of minor to major diameters
 106-110  F5.3  ---      e_b/a    ? Error on ratio of minor to major diameters
 112-116  F5.1  deg      PA       [0,180]? Position angle of galaxy
                                    (degrees from north through east)
 118-123  F6.2  mag      BMAG     ? Absolute blue magnitude
 125-131  F7.2  Mpc      Dist     ? Distance (Mpc)
 133-138  F6.2  Mpc      e_Dist   ? error on Distance (Mpc)
 140-143  F4.2  mag      e_Bmag   ? error on Apparent blue magnitude
 145-148  F4.2  mag      e_BMAG   ? error on Absolute blue magnitude

'object' (count 19): 
 - Name
 - Type
 - App_Mag_B
 - err_App_Mag_B
 - Abs_Mag_B
 - err_Abs_Mag_B
 - App_Mag_I
 - err_App_Mag_I
 - Abs_Mag_I
 - err_Abs_Mag_I
 - App_Mag_K
 - err_App_Mag_K
 - Maj_Diam(a)
 - err_Maj_Diam
 - Min_Diam(b)
 - err_Min_Diam
 - b/a
 - err_b/a
 - PA
'float64' (count 4): 
 - RA
 - Dec
 - Dist
 - err_Dist
'int64' (count 1): 
 - PGC

	Name	Dist	err_Dist	RA	Dec	Type	App_Mag_B
63638	PGC138606	34.347	5.152	3.18005	61.105202	5.0	16.32
68443	PGC166081	72.361	15.919	4.09247	71.469597	7.0	17.47
68880	PGC166081	24.847	3.727	4.09246	71.469470	NaN	17.39
130474	PGC138606	35.569	7.825	3.17891	61.113250	1.2	15.51
178917	6dFJ1705055-200214	112.653	24.784	17.08485	-20.037250	NaN	NaN
178919	6dFJ1704153-203840	117.694	25.893	17.07090	-20.644350	NaN	NaN
179298	6dFJ1705055-200214	112.653	24.784	17.08485	-20.037250	NaN	NaN
179620	6dFJ1704153-203840	116.194	25.563	17.07090	-20.644350	NaN	NaN

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1)

Data prepare

1. [x] load GWGC data

2. [x] check duplicates

3. [=] extract local galaxies within 30 Mpc and 3D view: 8,946 galaxies out of 183,327, present in 3D plot

4. spatial distribution and edge galaxies: present in distplot

5. check interesting features

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1)

Data prepare

1. [x] load GWGC data

2. [x] check duplicates

3. [x] extract local galaxies within 30 Mpc and 3D view

4. [=] spatial distribution and edge galaxies: present in distplot

   • D < 30 Mpc but D + $\Delta$D > 30 Mpc: 1,856 orange
   • D > 30 Mpc but D - $\Delta$D < 30 Mpc: 3,015 blue
   • be cautious with bin in histgrams
   • galaxy morphological type, numerical Hubble stage, not well explained in paper or references therein: => a future topic
     • [-4, -3]: E/S0 => GCpG.iMType==6
     • [0, 11]: S/Irr => GCpG.iMType==1
     • [-6, -4): E => GCpG.iMType==2
     • (-3, 0): S0 => GCpG.iMType==3
     • NaN: Unknown, 0
5. check interesting features

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1)

Data prepare

1. [x] load GWGC data

2. [x] check duplicates

3. [x] extract local galaxies within 30 Mpc and 3D view

4. [x] spatial distribution and edge galaxies

5. [=] check interesting features: missing data

   • statistics: present in pandas table
   • missing data visualization: present in msnoplot

	Name	RA	Dec	Dist	Type	Abs_Mag_B	Abs_Mag_I
count	8946	8946.000000	8946.000000	8946.000000	5979.000000	7877.000000	5606.000000
unique	8946	NaN	NaN	NaN	NaN	NaN	NaN
min	NaN	0.005940	-89.334590	0.010000	-5.000000	-22.590000	-25.520000
max	NaN	23.996680	86.131800	29.986000	10.000000	-4.970000	-1.680000
mean	NaN	10.973250	3.455383	19.011495	4.524034	-16.210734	-17.158776
std	NaN	5.277991	34.889882	6.616150	4.707900	2.451796	2.627928

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1): 8,946 galaxies within 30 Mpc out of 183,327 in GWGC

Harris catalog for GCpG model (sec 2.2.1)

GCpG models (sec 2.2)

Modeling GC populations (sec 2.3)

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

• Harris et al. (2013)
• data source

Data prepare

1. load GCpG data
2. check Morphology Type

3. check duplicates and NaNs

4. check interested features

Data explore

Description

0   1- 14  A14   ---        Name    Galaxy identifier
   16- 27  A12   ---        OName   Other galaxy identifier
   29- 38  F10.7 h          RAhour  ? Right Ascension in decimal hours (J2000)
   40- 50  F11.7 deg        DEdeg   ? Declination in decimal degrees (J2000)
4  52- 56  A5    ---        MType   Morphology type
5  58- 63  F6.2  Mpc        Dist    ? NED distance; see Section 2
   65- 69  F5.2  Mpc        e_Dist  ? Uncertainty in Dist
   71- 83  A13   ---        n_Dist  Method used to determine Dist
   85- 89  F5.3  mag        Av      NED foreground V band extinction
9  91- 97  F7.3  mag        VMag    [-25/-11]? Absolute visual magnitude (1)
   99-104  F6.3  mag        e_VMag  [0.2/1.5]? Uncertainty in VMag
11 106-115  F10.3 mag        KMag    [-28/-14]? Absolute K band magnitude (2)
   117-125  F9.3  mag        e_KMag  [0.1/0.3]? Uncertainty in KMag
       127  A1    ---        l_Ngc   [>] Limit flag on Ngc
14 128-134  F7.1  ---        Ngc     [0/32500] Number of Globular Clusters (NGC)
   136-142  F7.1  ---        e_Ngc   [0/20000] Uncertainty in Ngc
   144-152  A9    ---        r_Ngc   Reference(s) for Ngc (see refs.dat file)
17 154-159  F6.2  km/s       sigma   [10/414]? Stellar velocity dispersion (3)
   161-165  F5.2  km/s      e_sigma  ? Uncertainty in sigma
19 167-171  F5.2  kpc        Reff    [0.1/55]? Effective radius (4)
   173-177  F5.2  kpc        e_Reff  ? Uncertainty in Reff
21 179-184  F6.3  [Msun]     logMd   [7.6/12.8]? Log of dynamical mass
   186-191  F6.3  [Msun]    e_logMd  ? Uncertainty in logMd
23 193-197  F5.2  [Msun]     logMGC  ? Log of total globular cluster mass (5)
   199-203  F5.2  [Msun]   e_logMGC  ? Uncertainty in logMGC
25 205-209  F5.2  [Msun]     logMB   ? Log of central supermassive black hole mass(6)
   211-215  F5.2  [Msun]    E_logMB  ? Upper uncertainty in logMB
   217-221  F5.2  [Msun]    e_logMB  ? Lower uncertainty in logMB

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

1. [=] load GCpG data
2. [=] check Morphology Type: MType: str => to numeric, & parse in Harris manner, present in barplot

3. [=] check duplicates and NaNs: explain pandas table, 418 galaxy enteries

   • 0: Nan in [VMag, KMag, Dist], MilkyWay => drop
   • 356: Nan in [VMag, KMag], A1689-BCG, 770 Mpc => drop
   • 227,228: completely duplicated => drop 228
   • 272,273: different data reduction => drop 272(higher uncertainty in data)
   • 138: NaN in logMGC, 3 in Ngc => data is dirty, check in 4-A-b

4. check interested features

Data explore

Our catalog contains 248 ellipticals, 93 S0's, and 81 spirals or irregulars, (Harris et. al. 2013).

	Name	RAhour	DEdeg	Dist	e_Dist	VMag	e_VMag	KMag	logMGC	Ngc	r_Ngc	sigma	Reff	logMd	logMB
0	MilkyWay	NaN	NaN	NaN	NaN	-21.30	0.3	NaN	7.66	160.0	44	105.0	0.70	9.856	6.61
356	A1689-BCG	13.191528	-1.338056	790.00	40.00	NaN	NaN	NaN	10.10	30000.0	66b	NaN	NaN	NaN	NaN
227	NGC4417	12.447387	9.584243	16.03	0.53	-20.00	0.2	-22.860	7.24	72.0	77	135.2	1.22	10.318	NaN
228	NGC4417	12.447387	9.584243	16.03	0.53	-20.00	0.2	-22.860	7.24	72.0	77	135.2	1.22	10.318	NaN
272	VCC-1386	12.530939	12.656659	16.00	2.00	-17.34	0.2	NaN	6.67	26.6	67	20.5	1.80	8.848	NaN
273	VCC-1386	12.530939	12.656659	16.10	1.13	-16.90	0.2	NaN	6.64	26.0	24	NaN	NaN	NaN	NaN
138	NGC2915	9.436528	-76.626389	4.06	0.20	-16.12	0.2	-18.298	NaN	3.0	66a	NaN	0.31	NaN	NaN

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

1. [x] load GCpG data
2. [x] check Morphology Type

3. [x] check duplicates and NaNs

4. [=] check interested features:

   1. [=] statistics, present in 2 pandas tables

      • tab1: var(Ngc(6:0)) significantly larger => add logNgc(6:-inf)
      • tab2: explain unreasonable value in Ngc, e_Ngc(0:3), logMGC(1:NaN), e_logMGC(2:0):
        • Ngc derived from logMGC => focus on Ngc/logNgc(412/418)
        • Ngc == 0: [7,51,87,88,271,304] => Ngc = 0.5 e_Ngc, avoid -inf temporately
        • e_Ngc == 0: [1,28,138] => e_Ngc = 0.5 Ngc, avoid -inf temporately
        • logMGC, e_logMGC == NaN: [138] => linear interpolate NaN logMGC from logNgc(412))
        • e_logMGC ==0: [1,28] => leave alone

   2. visualization

Data explore

	Name	VMag	KMag	Ngc	e_Ngc	r_Ngc	sigma	Reff	logMd	logMGC	Dist	logMB
count	418	418.000000	344.000000	418.000000	418.000000	418	271.000000	340.000000	253.000000	417.000000	418.000000	64.000000
unique	418	NaN	NaN	NaN	NaN	106	NaN	NaN	NaN	NaN	NaN	NaN
min	NaN	-24.190000	-27.080000	0.000000	0.000000	NaN	10.400000	0.130000	7.693000	4.880000	0.030000	0.000000
max	NaN	-11.170000	-14.510000	32500.000000	13000.000000	NaN	414.000000	55.000000	12.726000	10.120000	375.000000	10.320000
mean	NaN	-19.127943	-22.583840	1512.699761	351.961962	NaN	168.750923	4.287588	10.727692	7.369305	34.758971	5.933750
std	NaN	2.799122	2.659367	3957.436127	1075.182192	NaN	93.365634	6.092928	0.969581	1.229806	46.445852	3.953719

	Name	Ngc	e_Ngc	logMGC	e_logMGC
7	NGC221	0.0	1.0	6.26	0.10
51	FCC-0059	0.0	4.0	6.21	0.19
87	FCC-242	0.0	3.3	5.51	0.30
88	FCC-246	0.0	3.4	5.51	0.30
138	NGC2915	3.0	0.0	NaN	NaN
271	VCC-1363	0.0	3.0	6.19	0.24
304	VCC-1714	0.0	4.5	6.19	0.16

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

1. [x] load GCpG data
2. [x] check Morphology Type

3. [x] check duplicates and NaNs

4. [=] check interested features

   1. [x] statistics, present in 2 pandas tables

   2. [=] visualization

      1. logNgc: present KDE in distplot
      2. data completeness: represent in barplot
      3. spatial sparseness of NaN data: present in msnoplot

Data explore

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [=] explore correlations for data type in float: present in pairplots with heatmap

   • properties:
     • 'Dist', 'e_Dist', 'Av', 'VMag', 'e_VMag', 'KMag', 'e_KMag', 'Ngc', 'e_Ngc', 'sigma', 'e_sigma', 'Reff', 'e_Reff', 'logMd', 'e_logMd', 'logMGC', 'e_logMGC','logMB', 'E_logMB', 'e_logMB', 'logNgc'
   • correlated =>
     • xxx & e_xxx: intuitive, not interesting
     • Av & Dist: Dist derived from NED Av => focus on Dist (more intuitive)
     • VMag & KMag(74:NaN): intuitive, LR in 2-A
   • probably correlated =>
     • Dist & Ngc(6:0) & Reff(78:NaN)
     • VMag & KMag(74:NaN) & sigma(147:NaN) & logMd(165:NaN) & logMGC(1:NaN) & logNgc(6:-inf)

2. check correlated pairs [and interpolate values]

3. correlation ranking

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [=] check correlated pairs [and interpolate values]

   1. [=] VMag(418) & KMag(344+74): LR interpolate => KMag(74:NaN), present in errorbar & OLS LR summary

   2. Dist & Ngc(6:0) & Reff(78:NaN)

   3. VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)

3. correlation ranking

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.999    
Dependent Variable: y                AIC:                803.0247 
Date:               2018-04-27 12:47 BIC:                807.0602 
No. Observations:   418              Log-Likelihood:     -400.51  
Df Model:           1                F-statistic:        5.005e+05
Df Residuals:       417              Prob (F-statistic): 0.00     
R-squared:          0.999            Scale:              0.39886  
---------------------------------------------------------------------
         Coef.     Std.Err.       t        P>|t|     [0.025    0.975]
---------------------------------------------------------------------
x1       1.1304      0.0016    707.4353    0.0000    1.1273    1.1336
------------------------------------------------------------------
Omnibus:              176.239      Durbin-Watson:         1.879   
Prob(Omnibus):        0.000        Jarque-Bera (JB):      1964.668
Skew:                 1.479        Prob(JB):              0.000   
Kurtosis:             13.201       Condition No.:         1       
==================================================================

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [=] check correlated pairs [and interpolate values]

   1. [x] VMag(418) & KMag(344+74)

   2. [=]Dist & Ngc(6:0) & Reff(78:NaN): no significant correlation => leave alone, present in pairplot

   3. VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)

3. correlation ranking

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [x] check correlated pairs [and interpolate values]

   1. [x] VMag(418) & KMag(344+74)

   2. [x] Dist & Ngc(6:0) & Reff(78:NaN)

   3. [=] VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)

      1. [=] logNgc(6:-inf) & logMGC(1:NaN): Ngc(6:0) deduct from logMGC (Harris et al. 2011), present in errorbar & OLS LR summary

         • => logNgc(412-1) LR interpolate logMGC(1:NaN)
         • => logMGC(418-1-6) LR interpolate Ngc/logNgc(6:0/-inf)

      2. VMag/KMag (418/344+74) & logNgc/logMGC (412/417)

      3. logMd(253) & sigma(271) & logNgc/logMGC (412/417)

3. correlation ranking

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.892    
Dependent Variable: y                AIC:                954.0297 
Date:               2018-04-27 12:47 BIC:                958.0651 
No. Observations:   418              Log-Likelihood:     -476.01  
Df Model:           1                F-statistic:        3437.    
Df Residuals:       417              Prob (F-statistic): 1.75e-203
R-squared:          0.892            Scale:              0.57241  
---------------------------------------------------------------------
          Coef.     Std.Err.       t       P>|t|     [0.025    0.975]
---------------------------------------------------------------------
x1        0.2906      0.0050    58.6292    0.0000    0.2808    0.3003
------------------------------------------------------------------
Omnibus:               40.119       Durbin-Watson:          1.430 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       18.367
Skew:                  0.316        Prob(JB):               0.000 
Kurtosis:              2.191        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [x] check correlated pairs [and interpolate values]

   1. [x] VMag(418) & KMag(344+74)

   2. [x] Dist & Ngc(412+6) & Reff(78:NaN)

   3. [=] VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)

      1. [x] logNgc(412+6) & logMGC(417+1)

      2. [=] VMag/KMag (418/344+74) & logNgc/logMGC (412/417): all complete

         • => VMag(original) vs logNgc (more intuitive)
         • => Luminosity model in sec 2.2.1, present in pairplot

      3. logMd(253) & sigma(271) & logNgc/logMGC (412/417)

3. correlation ranking

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [x] check correlated pairs [and interpolate values]

   1. [x] VMag(418) & KMag(344+74)

   2. [x] Dist & Ngc(6:0) & Reff(78:NaN)

   3. [=] VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417+1)/logNgc(412+6)

      1. [x] logNgc(412+6) & logMGC(417+1)

      2. [x] VMag/KMag (418/344+74) & logNgc/logMGC (412+6/417+1)

      3. [=] logMd(253) & sigma(271) & logNgc/logMGC (412+6/417+1)

         • => logMd(253) = f(sigma(271),Reff(340)) vs logNgc(412+6):logMd(253) limiting term
         • => Dynamical mass model in sec 2.2.2, present in pairplot

3. correlation ranking

Populating GCs in the local universe (chap 2)

Harris catalog for GCpG model (sec 2.2.1)

Data prepare

Data explore

1. [x] explore correlations for data type in float

2. [x] check correlated pairs [and interpolate values]

   1. [x] VMag & KMag

   2. [x] Dist & Ngc & Reff

   3. [x] VMag(KMag) & logMd (sigma) & logMGC(logNgc)

3. [=] correlation ranking: Random Forest with feature importance, explain in heatmap

   • VMag/KMag(344+74) vs logNgc(412+6)/logMGC(417+1)
   • impute median for NaN: => KMag & logMGC > KMag & logNgc > VMag & logNgc
   • impute mean for NaN: => KMag & logNgc > KMag & logMGC > VMag & logMGC
   • impute most frequent for NaN: => KMag & logNgc > KMag & logMGC > VMag & logMGC
   • 'normalize' by data availibility: 253/418(logMd) subset => KMag & Ngc > logMd & logMGC > VMag & logMGC, max correlation score drops
   • => possible selection effect
   • => overfitting in interpolation
   • => debating GCpG nature
   • (color-blind/friendly seaborn colors and matplotlib)

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1): 8,946 galaxies within 30 Mpc

Harris catalog for GCpG model (sec 2.2.1): linear model with Luminosity / Dynamical mass

GCpG models (sec 2.2)

Modeling GC populations (sec 2.3)

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: logMd vs logMGC/logNgc

1. [=] dynamical mass model logMGC/logNgc $=\mathscr{LR}$(logMd)

   1. [=] scatter plot with LR: strong correlation, present in scatter + regplot

   2. OLS LR summary

2. luminosity model $\mathscr{LR}$(VMag/KMag)

3. revisit with errorbar

GC SN model

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: logMd vs logMGC/logNgc

1. [=] dynamical mass model logMGC/logNgc $=\mathscr{LR}$(logMd)

   1. [x] scatter plot with LR

   2. [=] OLS LR summary: for logMd(253): logMGC(0.996) > logNgc(0.923)

2. luminosity model $\mathscr{LR}$(VMag/KMag)

3. revisit with errorbar

GC SN model

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.923    
Dependent Variable: y                AIC:                566.4509 
Date:               2018-04-27 12:48 BIC:                569.9842 
No. Observations:   253              Log-Likelihood:     -282.23  
Df Model:           1                F-statistic:        3037.    
Df Residuals:       252              Prob (F-statistic): 1.41e-142
R-squared:          0.923            Scale:              0.54723  
---------------------------------------------------------------------
          Coef.     Std.Err.       t       P>|t|     [0.025    0.975]
---------------------------------------------------------------------
x1        0.2379      0.0043    55.1067    0.0000    0.2294    0.2464
------------------------------------------------------------------
Omnibus:              10.559        Durbin-Watson:           1.338
Prob(Omnibus):        0.005         Jarque-Bera (JB):        5.680
Skew:                 0.159         Prob(JB):                0.058
Kurtosis:             2.338         Condition No.:           1    
==================================================================

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.996    
Dependent Variable: y                AIC:                340.7403 
Date:               2018-04-27 12:48 BIC:                344.2737 
No. Observations:   253              Log-Likelihood:     -169.37  
Df Model:           1                F-statistic:        7.156e+04
Df Residuals:       252              Prob (F-statistic): 2.49e-311
R-squared:          0.996            Scale:              0.22424  
---------------------------------------------------------------------
         Coef.     Std.Err.       t        P>|t|     [0.025    0.975]
---------------------------------------------------------------------
x1       0.7394      0.0028    267.5077    0.0000    0.7339    0.7448
------------------------------------------------------------------
Omnibus:              5.236         Durbin-Watson:           1.515
Prob(Omnibus):        0.073         Jarque-Bera (JB):        7.433
Skew:                 -0.017        Prob(JB):                0.024
Kurtosis:             3.839         Condition No.:           1    
==================================================================

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: VMag/KMag vs logNgc

1. [x] dynamical mass model $\mathscr{LR}$(logMd)

2. [=] luminosity model logNgc $=\mathscr{LR}$(VMag/KMag)

   1. [=] scatter plot with LR: strong correlation, present in scatter + regplot

   2. OLS LR summary

3. revisit with errorbar

GC SN model

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: VMag/KMag vs logNgc

1. [x] dynamical mass model $\mathscr{LR}$(logMd)

2. [x] luminosity model logNgc=$\mathscr{LR}$(VMag/KMag)

   1. [x] scatter plot with LR

   2. [=] OLS LR summary

      • VMag(0.866) < KMag(0.869) < logMd(0.923/0.996)
      • recall logMd(253/418), KMag(344/418) interpolated from VMag
      • => might be selection effect: logMd requires good photometry data
      • => overfitting in interpolation of KMag

3. revisit

GC SN model

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.866    
Dependent Variable: y                AIC:                1043.0025
Date:               2018-04-27 12:48 BIC:                1047.0379
No. Observations:   418              Log-Likelihood:     -520.50  
Df Model:           1                F-statistic:        2698.    
Df Residuals:       417              Prob (F-statistic): 3.34e-184
R-squared:          0.866            Scale:              0.70819  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.1106      0.0021    -51.9461    0.0000    -0.1148    -0.1064
------------------------------------------------------------------
Omnibus:               24.883       Durbin-Watson:          1.467 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.434
Skew:                  0.434        Prob(JB):               0.000 
Kurtosis:              2.397        Condition No.:          1     
==================================================================

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.870    
Dependent Variable: y                AIC:                1030.4039
Date:               2018-04-27 12:48 BIC:                1034.4393
No. Observations:   418              Log-Likelihood:     -514.20  
Df Model:           1                F-statistic:        2794.    
Df Residuals:       417              Prob (F-statistic): 6.21e-187
R-squared:          0.870            Scale:              0.68716  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.0980      0.0019    -52.8557    0.0000    -0.1017    -0.0944
------------------------------------------------------------------
Omnibus:               22.487       Durbin-Watson:          1.448 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.376
Skew:                  0.453        Prob(JB):               0.000 
Kurtosis:              2.458        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models:

1. [x] dynamical mass model logMGC/logNgc=$\mathscr{LR}$(logMd)

2. [x] luminosity model logNgc=$\mathscr{LR}$(VMag/KMag)

3. [=] revisit with errorbar, present in 2 errorbars

   1. [=] dynamical mass model: limited by data completeness (253/418)
   2. luminosity model

GC SN model

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.870    
Dependent Variable: y                AIC:                1030.4039
Date:               2018-04-27 12:48 BIC:                1034.4393
No. Observations:   418              Log-Likelihood:     -514.20  
Df Model:           1                F-statistic:        2794.    
Df Residuals:       417              Prob (F-statistic): 6.21e-187
R-squared:          0.870            Scale:              0.68716  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.0980      0.0019    -52.8557    0.0000    -0.1017    -0.0944
------------------------------------------------------------------
Omnibus:               22.487       Durbin-Watson:          1.448 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.376
Skew:                  0.453        Prob(JB):               0.000 
Kurtosis:              2.458        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: logMd vs logMGC/logNgc and VMag/KMag vs logNgc

1. [x] dynamical mass model logMGC/logNgc=$\mathscr{LR}$(logMd)

2. [x] luminosity model logNgc=$\mathscr{LR}$(VMag/KMag)

3. [=] revisit with errorbar, present in 2 errorbars

   • [x] dynamical mass model (253/418)
   • [=] luminosity model: no better with huge uncertainties

GC SN model

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.870    
Dependent Variable: y                AIC:                1030.4039
Date:               2018-04-27 12:48 BIC:                1034.4393
No. Observations:   418              Log-Likelihood:     -514.20  
Df Model:           1                F-statistic:        2794.    
Df Residuals:       417              Prob (F-statistic): 6.21e-187
R-squared:          0.870            Scale:              0.68716  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.0980      0.0019    -52.8557    0.0000    -0.1017    -0.0944
------------------------------------------------------------------
Omnibus:               22.487       Durbin-Watson:          1.448 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.376
Skew:                  0.453        Prob(JB):               0.000 
Kurtosis:              2.458        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: bothare messy and trustless

GC SN model

1. [=] the $N_\text{GC}$ per unit of parent galaxy luminosity VMag, normalized to $M_V$ = −15, present in scatter
   • $S_N \equiv N_\text{GC} \times 10^{0.4(M_V^T + 15)}$
   • $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$

2. [=] LOWESS regression for $S_N$ vs VMag, present in plot

3. error comparison

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.870    
Dependent Variable: y                AIC:                1030.4039
Date:               2018-04-27 12:48 BIC:                1034.4393
No. Observations:   418              Log-Likelihood:     -514.20  
Df Model:           1                F-statistic:        2794.    
Df Residuals:       417              Prob (F-statistic): 6.21e-187
R-squared:          0.870            Scale:              0.68716  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.0980      0.0019    -52.8557    0.0000    -0.1017    -0.0944
------------------------------------------------------------------
Omnibus:               22.487       Durbin-Watson:          1.448 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.376
Skew:                  0.453        Prob(JB):               0.000 
Kurtosis:              2.458        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

GCpG models (sec 2.2)

Linear models: bothare messy and trustless

GC SN model

1. [x] the $N_\text{GC}$ per unit of parent galaxy luminosity VMag
   • $S_N \equiv N_\text{GC} \times 10^{0.4(M_V^T + 15)}$
   • $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$

2. [x] LOWESS regression for $S_N$ vs VMag

3. [=] error comparison, zoom in view, justificationin errorbar

   • => error suppressed on high luminosity end: a few noisy outliners less-weighted
   • => fair performance on low luminosity end: low influence as the base $M_V$ will be low
   • => well represented in the most range
   • => truncated by VMag range [-11.17, -24.19]: lower-bound

                 Results: Ordinary least squares
==================================================================
Model:              OLS              Adj. R-squared:     0.870    
Dependent Variable: y                AIC:                1030.4039
Date:               2018-04-27 12:48 BIC:                1034.4393
No. Observations:   418              Log-Likelihood:     -514.20  
Df Model:           1                F-statistic:        2794.    
Df Residuals:       417              Prob (F-statistic): 6.21e-187
R-squared:          0.870            Scale:              0.68716  
---------------------------------------------------------------------
       Coef.     Std.Err.       t        P>|t|      [0.025     0.975]
---------------------------------------------------------------------
x1    -0.0980      0.0019    -52.8557    0.0000    -0.1017    -0.0944
------------------------------------------------------------------
Omnibus:               22.487       Durbin-Watson:          1.448 
Prob(Omnibus):         0.000        Jarque-Bera (JB):       19.376
Skew:                  0.453        Prob(JB):               0.000 
Kurtosis:              2.458        Condition No.:          1     
==================================================================

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1): 8,946 galaxies within 30 Mpc

Harris catalog for GCpG model (sec 2.2.1): linear model with Luminosity / Dynamical mass

GCpG models (sec 2.2): a modest GC SN model

Modeling GC populations (sec 2.3)

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [=] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [=] GWGC luminosities, Abs_Mag_B/I(7,877/5,606), App_Mag_B/I/K(7,877/5,606/2,995), and err_Abs/App_Mag_B/I/K:

      • count statistics, present in pandas table
      • completeness, msnoplot, lots of NaNs

   2. explore correlations

   3. fit conversion constant

   4. choose most probable value

2. revert to Ngc

3. GC distribution

	App_Mag_B	err_App_Mag_B	Abs_Mag_B	err_Abs_Mag_B	App_Mag_I	err_App_Mag_I	Abs_Mag_I	err_Abs_Mag_I	App_Mag_K	err_App_Mag_K
count	7877.000000	7877.000000	7877.000000	7877.000000	5606.000000	5606.000000	5606.000000	0.0	2995.000000	2287.00000
min	-4.600000	0.040000	-22.590000	0.070000	5.730000	0.030000	-25.520000	NaN	7.271000	0.02000
max	25.260000	3.020000	-4.970000	3.020000	30.440000	6.880000	-1.680000	NaN	15.978000	0.29000
mean	14.981126	0.377188	-16.210734	0.387227	14.158673	0.174930	-17.158776	NaN	12.968526	0.10208
std	2.280469	0.176294	2.451796	0.175116	2.463852	0.317297	2.627928	NaN	1.594678	0.04896
NaNs	1069.000000	1069.000000	1069.000000	1069.000000	3340.000000	3340.000000	3340.000000	8946.0	5951.000000	6659.00000

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [=] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [x] GWGC luminosities

   2. [=] explore correlations, explain in pairplots with heatmap

      • correlated
        • => xxx & err_xxx: intuitive, not interesting
        • => abs_x & app_x: b'cos $app\_x-abs\_X=5\log_{10}(d)-5$
        • => err_Abs_x & err_App_x: of course, pass
      • all bands are monochromatic (AB magnitude system)
      • => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $

   3. fit conversion constant

   4. choose most probable value

2. revert to Ngc

3. GC distribution

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [=] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [x] GWGC luminosities

   2. [x] explore correlations

      • => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $

   3. [=] fit conversion constant, 197 galaxies in both catalogs, inspect in mag/color scatter

      • [Abs_Mag_B (7,877), err_Abs_Mag_B (7,877)] > [Abs_Mag_I (5,606), err_Abs_Mag_I (0)]: => prefer B over I for Ngc estimation
      • B/I in GWGC (197/148) vs V/K in GCSN=> almost homogenous, prefer B for fit
      • B/I in GWGC (197/148) vs V-K in GCSN=> color/Mcat didn't help (unequal axis)

   4. choose most probable value

2. revert to Ngc

3. GC distribution

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [=] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [x] GWGC luminosities

   2. [x] explore correlations

      • => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $

   3. [x] fit conversion constant: B/I in GWGC (197/148) vs V/K in GC SN on 197 shared galaxies

   4. [=] choose most probable value, based on kdeplot

      • =>$\mathrm{VMag}_\max = -20.515828946699045$
      • =>$\mathrm{BMag}_\max = -19.627615047946385$
      • =>$\mathrm{IMag}_\max = -21.008287606264027$

2. revert to Ngc

3. GC distribution

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [x] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [x] GWGC luminosities

   2. [x] explore correlations

      • => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $

   3. [x] fit conversion constant: B/I in GWGC (197/148) vs V/K in GC SN on 197 shared galaxies

   4. [x] choose most probable value, based on kdeplot

      • =>$\mathrm{VMag}_\max = -20.515828946699045$
      • =>$\mathrm{BMag}_\max = -19.627615047946385$
      • =>$\mathrm{IMag}_\max = -21.008287606264027$

2. [=] revert to Ngc:

   • convert BMag/IMag in GWGC to VMag in GCSN: B to V (7,877) + I to V (extra 32) = 7,909/8,946
   • apply $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$
   • result statistics and 5 'Interpolated Galaxy Samples', present in pandas table
     • => lower-bound GC Population is 658,910, over 8,946 galaxies within 30 Mpc
     • => 1,037 (11.59%) GWGC galaxy not in count
     • => 130 (1.45%) extra galaxy not counted due to GC SN truncation
     • => lowess fit is modest

3. GC distribution

	Name	RA	Dec	VMag	Dist	Abs_Mag_B	Abs_Mag_I	Ngc	e_Ngc	iMType
count	8946	8946.000000	8946.000000	7909.000000	8946.000000	7877.000000	5606.000000	8946.000000	8946.000000	8946.000000
min	NaN	0.005940	-89.334590	-23.478214	0.010000	-22.590000	-25.520000	0.000000	0.000000	0.000000
max	NaN	23.996680	86.131800	-5.858214	29.986000	-4.970000	-1.680000	11608.000000	7187.308776	6.000000
mean	NaN	10.973250	3.455383	-17.094336	19.011495	-16.210734	-17.158776	74.085848	46.053841	0.916387
std	NaN	5.277991	34.889882	2.452026	6.616150	2.451796	2.627928	262.963677	190.324592	0.936047
168135	SDSSJ142852.74+123454.0	14.481330	12.581660	NaN	28.958000	NaN	NaN	0.000000	0.000000	0.000000
168242	SDSSJ144213.99+333517.4	14.703880	33.588320	NaN	24.528000	NaN	NaN	0.000000	0.000000	0.000000
146627	SDSSJ080937.39+413526.6	8.160380	41.590710	-14.168214	11.625000	-13.280000	-12.870000	5.000000	2.037453	0.000000
24031	UGC06251	11.223940	53.595040	-17.718214	18.450000	-16.830000	NaN	25.000000	7.388049	1.000000
171592	SDSSJ151605.06+561614.7	15.268080	56.270820	-16.628214	9.875000	-15.740000	-16.790000	14.000000	3.364954	0.000000

Populating GCs in the local universe (chap 2)

Modeling GC populations (sec 2.3)

1. [x] bandpass conversion: B/I/K in GWGC from White et al. (2011) to V/K in GCSN model from Harris et al. (2013)

   1. [x] GWGC luminosities

   2. [x] explore correlations

      • => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $

   3. [x] fit conversion constant: B/I in GWGC (197/148) vs V/K in GC SN on 197 shared galaxies

   4. [x] choose most probable value, based on kdeplot

      • =>$\mathrm{VMag}_\max = -20.515828946699045$
      • =>$\mathrm{BMag}_\max = -19.627615047946385$
      • =>$\mathrm{IMag}_\max = -21.008287606264027$

2. [x] revert to Ngc: 7,909/8,946 with Ngc

   • => lower-bound GC Population is 658,910, over 8,946 galaxies within 30 Mpc
   • => 1,037 (11.59%) GWGC galaxy not in count
   • => 130 (1.45%) extra galaxy not counted due to GC SN truncation
   • => lowess fit is modest

3. [=] GC distribution

   • present in all-sky plot

Populating GCs in the local universe (chap 2)

GWGC for galaxy abundance (sec 2.1): 8,946 galaxies within 30 Mpc

Harris catalog for GCpG model (sec 2.2.1): linear model can be improved

GCpG models (sec 2.2): a modest GC SN model

Modeling GC populations (sec 2.3): 658,910 GCs in 7,909 out of the galaxies within 30 Mpc

	Name	RA	Dec	VMag	Dist	Abs_Mag_B	Abs_Mag_I	Ngc	e_Ngc	iMType
count	8946	8946.000000	8946.000000	0.0	8946.000000	7877.000000	5606.000000	0.0	0.0	8946.000000
min	NaN	0.005940	-89.334590	NaN	0.010000	-22.590000	-25.520000	NaN	NaN	0.000000
max	NaN	23.996680	86.131800	NaN	29.986000	-4.970000	-1.680000	NaN	NaN	6.000000
mean	NaN	10.973250	3.455383	NaN	19.011495	-16.210734	-17.158776	NaN	NaN	0.916387
std	NaN	5.277991	34.889882	NaN	6.616150	2.451796	2.627928	NaN	NaN	0.936047
168135	SDSSJ142852.74+123454.0	14.481330	12.581660	NaN	28.958000	NaN	NaN	NaN	NaN	0.000000
168242	SDSSJ144213.99+333517.4	14.703880	33.588320	NaN	24.528000	NaN	NaN	NaN	NaN	0.000000
146627	SDSSJ080937.39+413526.6	8.160380	41.590710	NaN	11.625000	-13.280000	-12.870000	NaN	NaN	0.000000
24031	UGC06251	11.223940	53.595040	NaN	18.450000	-16.830000	NaN	NaN	NaN	1.000000
171592	SDSSJ151605.06+561614.7	15.268080	56.270820	NaN	9.875000	-15.740000	-16.790000	NaN	NaN	0.000000

Simulating GCs (chap 3 + chap 4)

Modeling GCs

1. Characteristics (3.1)

   • Monte Carlo method (4.1)

2. Stellar evolution (3.2)

   • SSE and BSE (4.2.1)

3. Stellar dynamics (3.3)

   • Fewbody code (4.2.2)

4. External environment (3.4)

   • Escape (4.2.3)

5. MOCCA (4.2)

   • Monte Carlo + SSE and BSE + Fewbody code + Escape

Simulating GCs (chap 3 + chap 4)

Sampling GCs (sec 4.3)

1. General setup

2. Variations of parameters

3. Age spread: the Hubble time is assumed to be 13.5 Gyr

   • The Ages of 55 Globular Clusters as Determined Using an Improved $\Delta V^\text{HB}_\text{TO}$ Method along with Color-Magnitude Diagram Constraints, and Their Implications for Broader Issues Vanden Burg et al. (2013)

4. GC simulations on TACC

   • $2\times5\times3\times3\times3 \times 10 = 324 \times 10 = 3,240$ Models
   • Node-hour: $150,000 + 90,000 = 240,000$
   • tmp file: over 100 TB

5. GClib LIBRARY: TO BE RELEASED, 652,196 -> 662,772

Prospects of Detections (sec 5.2)

BBH formation in GCs (sec 5.2.2)

1. formation channels
2. one example in simulation

3. Assembling BBH database

   • BBHlib: 87,365 over 3,240 GCs

   • LoGal: 7,909 galaxies with GCs within 30 Mpc

   • GClib: 662,772 GCs over 7,909 galaxies

   • => BBHdata: 17,883,760 BBHs

Relativistic Evolution (sec 5.2.3)

Detector Response for LISA (sec 6.2.1)

Localization (sec 6.2.2)

Assembling BBH database

• BBHlib: extracted BBHs from 3,240 GC simulations

  • 87,365 ejected BBHs
  • 11,095 ejected BHBs

• LoGal: galaxy within 30 Mpc

  • 7,909 with GCs
  • 1,037 has no measured luminosity
  • 8,946 total

• GClib: 662,772 GCs over 7,909 Galaxy

  • manipulated LoGal.Ngc with different GC age

  • assign random GC model

  • GClib model diversity:

    • 1,739 out of 662,772 are duplicated
    • only half of the 3,240 GC model have one duplicated entry

• => BBHdata: 17,883,760 BBHs

    ```
    for each GC in LoGal.Ngc:
        assign random GC model
        get galaxy info. from LoGal
        get BBH from BBHlib with GC model
        combine
    ```

BBHdata

host galaxy info	GC info	BBH info
1-5	6-7	8-14
GC, RA, DEC, Dist, VMag	Model, T_GC (13.5Gyr-Age)[Fe/H, f_b, ...]	T_eject, M1, M2, Seperation,Ecc, (BBHlib.Model=GClib.Model)

	T_eject	M1	M2	Seperation	Ecc	Model
0	0.0079	26.460	11.6190	36.1060	0.3173	187-7
1	0.0118	22.999	7.8955	222.4900	0.0000	187-7
2	0.0157	23.190	8.1896	20.0940	0.0978	187-7
3	0.0315	23.175	8.2358	275.1300	0.4285	187-7
4	0.0315	22.895	7.7428	91.2530	0.8577	187-7
5	0.0315	23.874	9.4294	173.7900	0.7756	187-7
6	0.0315	25.761	10.9290	868.3400	0.0000	187-7
7	0.0499	25.777	11.1890	793.2600	0.3369	187-7
8	0.4144	13.386	25.7790	78.5730	0.8045	187-7
9	0.7613	26.697	26.9370	185.3200	0.5458	187-7
10	0.7886	37.914	28.3050	331.4400	0.3656	187-7
11	0.8329	26.815	26.1680	191.4900	0.8724	187-7
12	1.0465	23.922	24.4750	51.2410	0.6847	187-7
13	1.2962	25.195	25.3110	217.5200	0.5197	187-7
14	1.5873	26.058	23.4010	299.2500	0.9600	187-7
15	1.6343	26.829	26.8670	738.9600	0.7871	187-7
16	2.1382	25.227	26.1970	203.5100	0.6862	187-7
17	2.2051	26.931	24.2770	94.9120	0.4887	187-7
18	2.2264	26.946	24.3750	334.5500	0.7257	187-7
19	2.2484	27.270	26.6270	317.5200	0.7703	187-7
20	2.6205	23.248	25.5000	567.8200	0.9503	187-7
21	2.9080	21.622	24.2440	306.3300	0.5750	187-7
22	2.9240	23.420	23.5020	623.8000	0.3999	187-7
23	2.9433	23.193	40.5950	249.4500	0.8743	187-7
24	3.0288	23.418	22.5560	321.2600	0.8533	187-7
25	3.1998	26.911	23.5720	706.2900	0.6283	187-7
26	3.3023	22.814	24.3720	117.9900	0.9995	187-7
27	3.5353	22.164	20.8490	175.1500	0.9920	187-7
28	4.0062	13.906	13.7430	9.3148	0.8166	187-7
29	4.1225	25.177	26.7480	131.3900	0.2675	187-7
30	4.1471	25.910	52.3300	326.6700	0.6479	187-7
31	4.2271	24.400	13.2930	12.6890	0.6893	187-7
32	4.2399	23.670	24.8380	566.1000	0.8899	187-7
33	4.3531	20.876	28.1520	428.8500	0.5484	187-7
34	5.4659	24.859	22.3790	292.1900	0.9335	187-7
35	5.4829	23.489	21.2020	133.5700	0.6346	187-7
37	5.8268	24.884	49.2470	383.3000	0.6541	187-7
38	6.0204	27.398	19.7820	188.6200	0.4226	187-7
39	6.3426	22.117	23.3930	79.8000	0.4630	187-7
40	6.6529	27.150	11.6090	81.0850	0.7587	187-7
41	7.0097	22.594	27.0070	120.4400	0.4160	187-7
42	7.8008	24.080	26.8740	114.6100	0.8300	187-7
43	7.8008	24.624	25.3410	16084.0000	0.8807	187-7
44	8.3006	13.503	18.4360	148.1200	0.6899	187-7
45	8.3461	20.889	26.7890	189.2700	0.4582	187-7
47	9.4491	13.858	27.8840	138.2300	0.6108	187-7
48	9.7201	22.506	22.7170	197.7200	0.8736	187-7
49	10.4762	13.017	12.7020	175.4100	0.8080	187-7
50	10.7644	27.078	23.0610	243.2300	0.5025	187-7
51	11.0249	11.821	13.9440	303.7100	0.7290	187-7
53	11.7761	13.669	11.8270	233.4800	0.8816	187-7

	T_eject	M1	M2	Seperation	Ecc
count	87364.000000	87364.000000	87364.000000	8.736400e+04	87364.000000
mean	1.822836	24.921859	24.182866	5.255124e+02	0.685132
std	2.452034	17.785691	19.599724	2.390284e+04	0.279524
min	0.005200	3.000100	3.014100	0.000000e+00	0.000000
25%	0.237200	19.756000	13.493000	4.714150e+01	0.518700
50%	0.818000	23.190000	22.680000	9.762050e+01	0.765700
75%	2.389600	26.121000	27.013500	2.093825e+02	0.917400
max	13.095300	2669.500000	820.870000	5.894200e+06	1.000000

	Galaxy	RA	Dec	Dist	VMag	Model	Age
133	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866

	Galaxy	RA	Dec	Dist	VMag	Age
count	662772.000000	662772.000000	662772.000000	662772.000000	662772.000000	662772.000000
mean	30603.041664	10.959227	-0.355427	21.170577	-20.534914	11.890659
std	27284.737734	5.445347	38.155540	5.926124	1.693741	0.864954
min	25.000000	0.007920	-89.334590	0.010000	-23.478214	8.000000
25%	15238.000000	7.451130	-28.805970	17.783000	-21.698214	11.307087
50%	27130.000000	11.850490	-0.480240	21.478000	-20.888214	11.968504
75%	35109.000000	13.284170	25.500910	25.351000	-19.788214	12.535433
max	183326.000000	23.996680	86.131800	29.986000	-11.178214	13.480315

	Galaxy	RA	Dec	Dist	VMag	Model	Age	T_eject	M1	M2	Seperation	Ecc
0	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0079	26.460	11.6190	36.1060	0.3173
1	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0118	22.999	7.8955	222.4900	0.0000
2	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0157	23.190	8.1896	20.0940	0.0978
3	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0315	23.175	8.2358	275.1300	0.4285
4	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0315	22.895	7.7428	91.2530	0.8577
5	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0315	23.874	9.4294	173.7900	0.7756
6	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0315	25.761	10.9290	868.3400	0.0000
7	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.0499	25.777	11.1890	793.2600	0.3369
8	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.4144	13.386	25.7790	78.5730	0.8045
9	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.7613	26.697	26.9370	185.3200	0.5458
10	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.7886	37.914	28.3050	331.4400	0.3656
11	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	0.8329	26.815	26.1680	191.4900	0.8724
12	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	1.0465	23.922	24.4750	51.2410	0.6847
13	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	1.2962	25.195	25.3110	217.5200	0.5197
14	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	1.5873	26.058	23.4010	299.2500	0.9600
15	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	1.6343	26.829	26.8670	738.9600	0.7871
16	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.1382	25.227	26.1970	203.5100	0.6862
17	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.2051	26.931	24.2770	94.9120	0.4887
18	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.2264	26.946	24.3750	334.5500	0.7257
19	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.2484	27.270	26.6270	317.5200	0.7703
20	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.6205	23.248	25.5000	567.8200	0.9503
21	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.9080	21.622	24.2440	306.3300	0.5750
22	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.9240	23.420	23.5020	623.8000	0.3999
23	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	2.9433	23.193	40.5950	249.4500	0.8743
24	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	3.0288	23.418	22.5560	321.2600	0.8533
25	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	3.1998	26.911	23.5720	706.2900	0.6283
26	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	3.3023	22.814	24.3720	117.9900	0.9995
27	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	3.5353	22.164	20.8490	175.1500	0.9920
28	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.0062	13.906	13.7430	9.3148	0.8166
29	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.1225	25.177	26.7480	131.3900	0.2675
30	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.1471	25.910	52.3300	326.6700	0.6479
31	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.2271	24.400	13.2930	12.6890	0.6893
32	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.2399	23.670	24.8380	566.1000	0.8899
33	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	4.3531	20.876	28.1520	428.8500	0.5484
34	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	5.4659	24.859	22.3790	292.1900	0.9335
35	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	5.4829	23.489	21.2020	133.5700	0.6346
36	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	5.8268	24.884	49.2470	383.3000	0.6541
37	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	6.0204	27.398	19.7820	188.6200	0.4226
38	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	6.3426	22.117	23.3930	79.8000	0.4630
39	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	6.6529	27.150	11.6090	81.0850	0.7587
40	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	7.0097	22.594	27.0070	120.4400	0.4160
41	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	7.8008	24.080	26.8740	114.6100	0.8300
42	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	7.8008	24.624	25.3410	16084.0000	0.8807
43	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	8.3006	13.503	18.4360	148.1200	0.6899
44	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	8.3461	20.889	26.7890	189.2700	0.4582
45	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	9.4491	13.858	27.8840	138.2300	0.6108
46	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	9.7201	22.506	22.7170	197.7200	0.8736
47	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	10.4762	13.017	12.7020	175.4100	0.8080
48	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	10.7644	27.078	23.0610	243.2300	0.5025
49	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	11.0249	11.821	13.9440	303.7100	0.7290
50	133	0.05414	16.14555	13.183	-20.568214	187-7	11.070866	11.7761	13.669	11.8270	233.4800	0.8816

	PGC	RA	Dec	Dist	VMag	Age	T_eject	M1	M2	Seperation	Ecc	Period
count	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	1.788376e+07	0.0
mean	3.059303e+04	1.095810e+01	-4.172814e-01	2.116666e+01	-2.053641e+01	1.189068e+01	1.823276e+00	2.492480e+01	2.418357e+01	5.184244e+02	6.851611e-01	NaN
std	2.726805e+04	5.447050e+00	3.812537e+01	5.926283e+00	1.693210e+00	8.648354e-01	2.451835e+00	1.776007e+01	1.958754e+01	2.311852e+04	2.795283e-01	NaN
min	2.500000e+01	7.920000e-03	-8.933459e+01	1.000000e-02	-2.347821e+01	8.000000e+00	5.200000e-03	3.000100e+00	3.014100e+00	0.000000e+00	0.000000e+00	NaN
25%	1.523800e+04	7.451130e+00	-2.880597e+01	1.778300e+01	-2.169821e+01	1.130709e+01	2.375000e-01	1.975600e+01	1.349300e+01	4.715300e+01	5.188000e-01	NaN
50%	2.713300e+04	1.185049e+01	-4.986800e-01	2.147800e+01	-2.088821e+01	1.196850e+01	8.189000e-01	2.319300e+01	2.268100e+01	9.761800e+01	7.658000e-01	NaN
75%	3.510900e+04	1.328468e+01	2.550091e+01	2.535100e+01	-1.978821e+01	1.253543e+01	2.389600e+00	2.612100e+01	2.700700e+01	2.093100e+02	9.175000e-01	NaN
max	1.833260e+05	2.399668e+01	8.613180e+01	2.998600e+01	-1.117821e+01	1.348031e+01	1.309530e+01	2.669500e+03	8.208700e+02	5.894200e+06	1.000000e+00	NaN

Prospects of Detections (sec 5.2)

BBH formation in GCs (sec 5.2.2)

Relativistic Evolution (sec 5.2.3)

• formulae: Peters (1964)

  • $\langle \frac{da}{dt} \rangle = -\frac{64}{5} \frac{G^3 m_1 m_2(m_1 + m_2)}{c^5 a^3 (1-e^2)^{7/2}} \left( 1+\frac{73}{24}e^2 + \frac{37}{96}e^4 \right)$ (5.25)
  • $\langle \frac{de}{dt} \rangle = -\frac{304}{15} e \frac{G^3 m_1 m_2(m_1 + m_2)}{c^5 a^4 (1-e^2)^{5/2}} \left(1 + \frac{121}{304}e^2 \right)$ (5.26)

• code: Parallelized from Benacquista (2004)

  • input: BBHdata
  • output: BBHdata with updated Separation, Ecc, Period

• Merger event rate (6.1.3)

  • $\dot{f} = k_0 f^{11/3}$ (6.1) where $k_0=3.68\times 10^{-6} \mathscr{M}^{5/3}$
  • $n = \int \frac{\eta}{k_0} f^{-11/3} df = \frac{\eta}{k_0}\frac{3}{8} f_\text{min}^{-8/3}$ and $f_\text{min} \simeq 2\times 10^{-4.4} $ Hz
  • =>$\eta = N \frac{8 k_0}{3} f_\text{min}^{8/3} \left(\frac{30 \text{ Mpc}}{1 \text{ Gpc}} \right)^3 \left( \frac{13.5 \text{ Gyr}}{1 \text{ yr}} \right) \simeq 587 \text{ Gpc}^{-3} \text{ yr}^{-1}$

Detector Response for LISA (sec 6.2.1)

Localization (sec 6.2.2)

Prospects of Detections (sec 5.2)

BBH formation in GCs (sec 5.2.2)

Relativistic Evolution (sec 5.2.3)

Detector Response for LISA (sec 6.2.1)

• based on Cutler (1998); Belczynski et al. (2010); Downing et al. (2011)
• Resulting GW signal
• Compare with spectrum

Localization (sec 6.2.2)

Prospects of Detections (sec 5.2)

BBH formation in GCs (sec 5.2.2)

Relativistic Evolution (sec 5.2.3)

Detector Response for LISA (sec 6.2.1)

Localization (sec 6.2.2)

• Forward modeling by Cornish & Rubbo (2003b); Rubbo et al.(2004)
• Kyu's simple model (Eq. 6.14 to E1. 6.19)

Thank You!

Future

Publication

• Belczynski, K., Askar, A., Arca-Sedda, M., et al. 2017, ArXiv e-prints, arXiv:1712.00632
• Localization of BBH in the Local Universe (in Preparation)
• A Data Science approach towards the local abundance of Globular Clusters (in Preparation)

Data Release

• Local Globular Cluster Library
• Local Binary Black Hole database

contact: contact@domij.info

github repository: LocalBBH

Resume: 03/24/2018 cache